Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds

摘要

We estimate the L-P-norm (2 <= p <= +infinity) of the restriction to a curve of the eigenfunctions of the Laplace-Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere, these estimates are sharp. If the curve has nonvanishing geodesic curvature, we can improve our results. All our estimates are shown to be optimal for the sphere. Moreover, we sketch their extension to higher dimensions.